Although it is impossible to gain eggs of legendary pokemon (with the Creation Trio and Manaphy the only two exceptions, and even then, you need a mystery gift event to get them) in the games, it IS shown that Legendaries can have young in the anime (as seen with a mother Lugia and her baby Silver). I think I know why Lugia is so rare - The Pokemon Gorebyss.
Think about this, okay? Lugia are found in water-filled areas, and are even shown in the anime to live in the oceans (as seen in Pokemon 2000, the Power of One). As Pokemon reproduce with eggs, regardless whether or not the species is mammalian.
This means a Lugia with an egg stay with it after breeding for three reasons:
1-To incubate the egg
2-To protect it from thieves and ward off predators
3-To prevent it from being carried by ocean currents.
Whilst she is keeping the egg protected, the mother's body must be kept relatively still.
Enter a school of Gorebyss, which as stated in their R/S/E Pokedex entries, that they suck the blood out of prey until their bodies gain vivid colours to show that they have had their fill of blood. This means these Pokemon are essentially like giant aquatic ticks or mosquitoes.
Since Lugia is essentially a sitting duck whilst an egg is in its incubation, this means a whole school of Gorebyss can get easy meals off of the legendary Pokemon. Combined with the fact that they cannot move very much so as not to lose their egg, they cannot groom themselves properly to get rid of these parasitic bloodsuckers, meaning they will not be able to prevent major levels of blood loss.
What's more, if the Lugia is able to forcibly remove these Gorebyss, the thin mouths that they have can cause damage to a vein or artery, not only causing an open wound to get potentially inflamed by many factors including salt-water in the ocean, but the smell of blood may attract predator Pokemon in the waters, such as Gyrados. And since Lugia(17'01") is shorter than Gyrados(21'04"), the Atrocious Pokemon could easily wrap itself around the Legendary, and bite it in the neck to kill it*.
After this ordeal, the now-unattended egg is at major risk from the cold water, smaller aquatic predators(such as Sharpedo), larger aquatic Pokemon (such as Wailord), and water currents forcing it into large rocks, shipwrecks, and other aquatic hazards.
*I know Gyrados cannot naturally learn Wrap or similar moves, but it's a minor technicality considering that Pokemon that do learn it are of a similar body shape. And besides, although in Pokedex entries it states that Gorebyss does suck bodily fluids out of its prey, it cannot naturally learn Leech Life.